This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler’s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See Complex Numbers – Geometric Viewpoint.
A number of recent puzzles have involved perspective views of objects. I had never really explored the idea of a perspective map in detail. So some of the properties associated with it always seemed a bit vague to me. I decided I would derive the mathematical equations for the perspective or projective map and see how its properties fell out from the equations. With this information in hand I then addressed some questions I had about the article “Dürer: Disguise, Distance, Disagreements, and Diagonals!” by Annalisa Crannell, Marc Frantz, and Fumiko Futamura concerning a controversy over Albrecht Dürer’s woodcut St. Jerome in His Study (1514). And finally, I read somewhere that a parabola under a perspective map becomes an ellipse, so I was able to show that as well. See the Perspective Map.
(Update 7/1/2019) Continue reading
Sabine Hossenfelder wrote an excellent blog posting about the growing awareness that outstanding scientific problems are not getting solved at the same rate as in the past. Her whole article is worth a read, as are all her postings, but this latest contained a mathematical statement that warranted justification. For scientists “How much working time starting today corresponds to, say, 40 years working time starting 100 years ago. Have a guess! Answer: About 14 months.” See Hossenfelder Stagnation Problem.
Years ago during one of my many excursions into the history of mathematics I wondered how Mercator used logarithms in his map projection (introduced in a 1569 map) when logarithms were not discovered by John Napier (1550-1617) and published in his book Mirifici Logarithmorum Canonis Descriptio until 1614, three years before his death in 1617. The mystery was solved when I read a 1958 book by D. W. Waters which said Edward Wright (1561-1615) in his 1599 book Certaine Errors in Navigation produced his “most important correction, his chart projection, now known as Mercator’s.” Wright did not use logarithms explicitly but rather implicitly through the summing of discrete secants of the latitude as scale factors. But what really caught my attention in the Waters book was this arresting footnote: “Wright explained his projection in terms of a bladder blown up inside a cylinder, a very good analogy.” This article recounts my exploration of this idea. See Mercator Projection Balloon.
One thing I have always been curious about, but never got around to investigating, is how hard is it to see that the Lambert Equal-Area Projection of a sphere onto a cylinder in fact preserves areas? This 2012 essay attempts to provide an answer. The essay was recently updated to provide a link to the fabulous Youtube site by Grant Sanderson at 3blue1brown. He shows a different way of looking at the problem also without explicitly resorting to calculus. All his videos are spectacular and provide unparalleled insights into mathematics. What I wouldn’t give to have had such videos available when I was studying math. How much more quickly would I have been able to learn. See Lambert Equal Area Projection.
Having immersed myself in studying Kepler’s discovery that the planetary orbits were ellipses, I was immediately aware of how the British mathematician, Katie Steckles, justified her technique to cut an elliptical pizza into equal slices in her video of 14 March 2017. In her video Katie makes the claim that the result of any affine transformation of the circular pizza cut into equal sectors will also be a set of equal area slices. I made an attempt to substantiate these remarks. See Cutting Elliptical Pizza.
I have long been fascinated by Newton’s proof of Kepler’s Equal Areas Law and wanted to write about it. Of course, others have as well, but I wanted to emphasize an aspect of the proof that supported my philosophy of mathematics.
Before I get to Newton, however, I wanted to discuss how Kepler himself justified this law, since his approach has a number of fascinating historical aspects to it. I have previously discussed Kepler’s ellipse and in the process of doing that research, I came across a number of articles about how Kepler arrived at his equal areas law. One notable result is that even though now we call the idea that a planet orbits the Sun in an elliptical path with the Sun at one focus, Kepler’s First Law, and the idea that the line from the Sun to the planet sweeps out equal areas in equal times, Kepler’s Second Law, Kepler actually discovered these laws in reverse order. See Kepler’s Equal Areas Law
I had been exploring how Kepler originally discovered his first two laws and became fascinated by what he did in his Astronomia Nova (1609), as presented by a number of researchers. Among the writers was A. E. L. Davis. She mentioned that the characterization of the ellipse that Kepler was using was the idea of a “compressed circle,” that is, a circle all of whose points were shrunk vertically by a constant amount towards a fixed diameter of the circle. I did not recall ever hearing this idea before and tried to track down its origin together with a proof — futilely, Davis’s references notwithstanding. I then tried to prove it myself. It was easy to do with analytic geometry. But in the spirit of the Kepler era (before the advent of Fermat’s and Descartes’s beginnings at fusing algebra and geometry) I tried to prove it solely within Euclid’s plane geometry. Some critical steps seemed to come from the great work of Apollonius of Perga (262-190 BC) on Conics. But for me a final elegant proof was not evident until 1822 when Dandelin employed his inscribed spheres. See Kepler’s Ellipse.
In the process of exploring the compressed circle idea I acquired an immense appreciation and regard for Kepler and his perseverance in the face of the dominant paradigm of his era, namely, the 2000 year old idea that the celestial motions were all based on the most perfect motion of all, that of circles. The kinds of extremely laborious calculations he went through (just prior to the invention of logarithms by John Napier) were daunting, especially considering the trials he was undergoing in his personal life (trying to survive the religious destruction between Catholics and Protestants, along with defending his mother against charges of witchcraft).
Years ago (1963) I got the paperback The Calculus:A Genetic Approach, by Otto Toeplitz, which presented the basic ideas of the differential and integral calculus from a historical point of view. One thing Toeplitz did at the end of his book that I had not seen in other texts was to show the equivalence of Kepler’s Laws and Newton’s Law of Gravity. (Since 1963 David Bressoud has developed this theme in his excellent 1991 text.) I thought I would try to emulate Toeplitz’s approach with more modern notation (vectors) and arguments in hopes of extracting the essential ideas from the clutter.
A by-product of this effort was to reveal strongly the different paths that physics and mathematics follow in understanding physical reality. The mystery is that the mathematics ends up describing the physics so well. I will return to this theme a number of times in other posts. See Kepler’s Laws and Newton’s Laws.
Probably the most satisfying article I have put together is a recent one on point set topology. An old friend of mine, who studied math and physics in college but ended up getting a doctorate in English, asked me, what was topology? Knowing that there were two main branches of topology (general or point set topology and algebraic topology), I chose to describe point set topology first, especially since it was what I was most familiar with and had worked with most in my graduate work.
The essay turned out to have a surprising structure more like a musical theme and variations. The theme was the geometric series. I found it to be a wonderful medium to show the evolution of ideas (acting as variations) from the early Greeks (Zeno’s Paradoxes) through the development of calculus, decimal expansions of real numbers, to power series, metric spaces, and finally general topologies.
There was an additional benefit to this series of transformations of an initial idea: one of the major aspects of true mathematics became evident, namely, the extension of an idea into new territories that reveal unexpected connections to other forms of mathematics. Treating complicated functions as points in a topological space was a wonderful idea developed over the end of the 19th and beginning of the 20th centuries and became the basis of the field of functional analysis. See Point Set Topology (revised).
(Update 6/3/2021) Slightly revised version.
I happened to review this article and noticed I made a mistake in my integration example. I have no idea what I was thinking at the time, so I corrected it. As I reviewed the rest of the article, I noticed a bunch of “typos” that would make the text confusing, so I corrected those as well. And finally I rephrased wording in a couple of places to try to make things clearer.